How do I get rid of dust?

Does anyone have any tips for cleaning a basement which is full of dust from the workers feet bringing in mud for 3 months and no one ever bothering to clean the floor before they piled their junk in it? It's a slow process but it HAS to be done... I'm motivated today after spending the night alone because John has a cold and said the dust was bothering him so he went and slept on the couch upstairs.
I figure I'll sweep up the majority of it and get the rest with a vacuum, and then mop with bleach and laundry detergent in a bucket of water. But does anyone have any suggestions for first getting rid of the dust and then keeping it down?

We need to run a dehumidifier in our basement (it was put in this fall, so the dehumidifier helps to dry out the concrete all the way) and the water we pour out every day is SO dirty. I think it is all dust. That might help, or an air purifier maybe? If you find any cheaper solutions, let me know!

I just went thru a kitchen remodel and the dust went thru the entire house. I plugged in all of the air purifiers I own, changed the furnace filter and just kept vacuuming. Use an attachment and vacuum down the walls. Once you get thru vacuuming, dust the furniture, book cases, shelves, picture frames, etc with something like pledge that when you run a dust rag over it, the rag will stick to the dust. Use window cleaner on any picture glass and mirror. Then vacuum again then wash the floors. Keep checking the furnace and air purifier filters and change them if they get dirty while you stir up the dust.

It really doesn't matter what cleaners you use - using bleach doesn't work any better than just plain soap and water.

The brunt of my renovation was over a few months ago and I'm still finding dust. Good luck getting thru all it!!

Vacuum, vacuum, vacuum, and I used a swiffer (but now they have the swiffer duster) on all the walls to get the dust off of them when we had our hardwoods put in. I love my Swiffer duster. I have respiratory problems, and it works well enough it doesn't get my coughing all stirred up like regular dusting cloths.

Also, when you are cleaning, run the fan on your central air unit while you are cleaning, and for a little while after (and leave yourself a note to put it back on automatic-guess why I mention that! lol). Oh, and if you don't already, invest in a really good filter for your air return for at least while you are trying to get rid of all the dust- it makes a big difference in how much dust does and doesn't get recirculated. Hope that helps!